Documentation:Monte Carlo Equilibration

Monte Carlo equilibration

Theory

We have a timeseries of N measurements obtained from a Monte Carlo simulation, i.e. $y_0,y_1,\cdots,y_{N-1}$.

Suppose $\bar{y}_i = \beta_0 + \beta_1 x_i$ (s.t. $i = 0, 1, \cdots, N-1$) is the least-squares best fitted line, we attempt to minimize $S = \sum_i (y_i - \bar{y}_i)^2$ w.r.t. $\beta_0$ and $\beta_1$.

$\frac{\partial S}{\partial \beta_0 } = 0$ , $\frac{\partial S}{\partial \beta_1 } = 0$ :

$\Rightarrow \left( \begin{array}{cc} N & \sum_i x_i \\ \sum_i x_i & \sum_i x_i^2 \end{array} \right) \left( \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right) = \left( \begin{array}{c} \sum_i y_i \\ \sum_i x_i y_i \end{array} \right)$

$\Rightarrow \left( \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right) = \frac{1}{N\sum_i x_i^2 - (\sum_i x_i)^2} \left( \begin{array}{cc} \sum_i x_i^2 & -\sum_i x_i \\ -\sum_i x_i & N \end{array} \right) \left( \begin{array}{c} \sum_i y_i \\ \sum_i x_i y_i \end{array} \right)$

$\Rightarrow \beta_1 = \frac{N \sum_i x_i y_i - \sum_i x_i \sum_i y_i}{N\sum_i x_i^2 - (\sum_i x_i)^2}$

Slope of best-fitted line

$\Rightarrow \beta_1 = \frac{\sum_i (x_i - \bar{x}_i)( y_i - \bar{y}_i) }{\sum_i (x_i - \bar{x}_i)^2} \,\,\,\,\, \left( = \frac{s_{xy}}{s_{xx}} \right)$

Error in slope of best-fitted line

$\mathrm{Var}({\beta_1}) = \mathrm{Var} \left( \frac{\sum_i (x_i - \bar{x}_i)( y_i - \bar{y}_i) }{\sum_i (x_i - \bar{x}_i)^2} \right) = \frac{\sum_i (x_i - \bar{x}_i)^2 \mathrm{Var}(y_i) }{(\sum_i (x_i - \bar{x}_i)^2)^2}$

Denoting $\mathrm{Var}(y_i) = \sigma_{y}^2$, we have:

$\Rightarrow \mathrm{Var}({\beta_1}) = \sigma_{\beta_1}^2 = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2}$

$\Rightarrow \mathrm{Var}({\beta_1}) = \sigma_{\beta_1}^2 = \frac{ 12 \, \sigma_y^2 }{ N(N^2 - 1) }$

Hypothesis testing

$\begin{array}{l} H_0 : \beta_1 = 0 \\ H_1 : \beta_1 \ne 0 \end{array}$

Using the standard z-test, we reject $H_0$ at confidence interval $\alpha$ if

$\left| \frac{\beta_1 - 0 }{ \sigma_{\beta_1} } \right| > z_{\frac{1-\alpha}{2}}$